aa r X i v : . [ h e p - ph ] D ec The Extreme Walking Behavior in a 331-TC Model
A. Doff ∗ Universidade Tecnol´ogica Federal do Paran´a -UTFPR - DAFIS Av Monteiro Lobato Km 04,84016-210, Ponta Grossa, PR, Brazil (Dated: May 18, 2018)
Abstract
It is quite possible that the Technicolor problems are related to the poorly known self-energyexpression, or the way chiral symmetry breaking (CSB) is realized in non-abelian gauge theories.Actually, the only known laboratory to test the CSB mechanism is QCD. The TC dynamics maybe quite different from the QCD , this fact has led to the walking TC proposal making the newstrong interaction almost conformal and changing appreciably its dynamical behavior. There aredifferent ways to obtain of extreme walking (or quasi-conformal) technicolor theories, in this paperwe propose an scheme to obtain this behavior based on an extension of the electroweak sector ofthe standard model, in the context of so called 331-TC model. ∗ Electronic address: [email protected] . INTRODUCTION The 125 GeV new resonance discovered at the LHC [1] has many of the characteristicsexpected for the Standard Model (SM) Higgs boson. If this particle is a composite or anelementary scalar boson is still an open question that probably will be answered in the nextLHC run. In recent papers the ATLAS and CMS Collaborations[2] reported an experimentalanomaly in diboson production with apparent excesses in
W W , W Z and ZZ channels andthis anomaly have inspired a number of theoretical papers proposing as an explanation theproduction of heavy weak bosons, W ′ and Z ′ .Thus, it becomes interesting to investigate the possibility of obtaining a light scalar bosonin the context of models which features contributions from new heavy weak bosons W ′ and Z ′ . In some extensions of the standard model (SM), as in the so called 3-3-1 models[3] SU (3) L ⊗ SU (3) c ⊗ U (1) X , new massive neutral and charged gauge bosons, Z ′ and V ± , arepredicted. The 3-3-1 model is the minimal gauge group that at the leptonic level admitscharged fermions and their antiparticles as members of the same multiplet, the predictions ofthe GW = SU (2) L ⊗ U (1) Y alternative models are leptoquark fermions with electric charges5 / / L = ±
2. The quantization ofelectric charge is inevitable in the G = 3 m U (1) X coupling constant becomes strong and the exotic quark T (charge 5 /
3) will form a U (1) X condensate breaking SU (3) L ⊗ U (1) X to the electroweak symmetry. This possibilitywas explored by us in the Ref.[6] assuming a model based on the gauge symmetry SU (2) T C ⊗ SU (3) L ⊗ SU (3) c ⊗ U (1) X , where the electroweak symmetry is broken dynamically by atechnifermion condensate, that is characterized by the SU (2) T C
Technicolor (TC) gaugegroup. The early technicolor models [8] suffered from problems like flavor changing neutralcurrents (FCNC) and contributions to the electroweak corrections not compatible with theexperimental data, as can be seen in the reviews of Ref.[9]. However, the TC dynamicsmay be quite different from the known strong interaction theory, i.e. QCD, this fact hasled to the walking TC proposal [10], which are theories where the incompatibility with the2xperimental data has been sol-ved, making the new strong interaction almost conformaland changing appreciably its dynamical behavior.We can obtain an almost conformal TC theory, when the fermions are in the fundamentalrepresentation, introducing a large number of TC fermions ( n T F ), leading to an almost zero β function and flat asymptotic coupling constant. The cost of such procedure may be a largeS parameter[11] incompatible with the high precision electroweak measurements. However,this problem can be solved by assuming that TC fermions are in other representationsthan the funda-mental[12] and an effective Lagrangian analysis indicates that such modelsalso imply in a light scalar Higgs boson [13]. This possibility was also investigated andconfirmed through the use of an effective potential for composite operators [14] and througha calculation involving the Bethe-Salpeter equation (BSE) for the scalar state [15].The reason for the existence of the different models (or different potentials) for a compos-ite scalar boson, is a consequence of our poor knowledge of the strongly interacting theories,that is reflected in the many choices of parameters in the effective potentials. The possibil-ity of obtaining a light composite scalar according to the approach discussed in Ref.[14], isthat this result is a direct consequence of extreme walking (or quasi-conformal) technicolortheories, where the asymptotic self-energy behavior is described by Irregular form of TCfermions[14, 15][27] Σ (0) ( p ) ∼ µ (cid:2) bg ( µ ) ln (cid:0) p /µ (cid:1)(cid:3) − γ . (1)In the Ref.[16] we considered the possibility of a light composite scalar boson arising frommass mixing between a relatively light and heavy scalar singlets from a see-saw mechanismexpected to occur in two-scale Technicolor (TC) models and we identified that, regardless ofthe approach used for generating a light composite scalar boson, the behavior exhibited byextreme walking technicolor theories, is the main feature needed to produce a light compositescalar boson compatible with the boson observed at the LHC.After this brief motivation of the importance of extreme walking behavior to generatea light composite scalar boson in TC models, in addition to possibility of 331-TC modelcontain the necessary requirements to explain the anomaly in diboson production, in thispaper we propose an scheme to obtain the quasi-conformal behavior based on an extensionof the electroweak sector of the standard model, 331-TC model ( SU ( N ) T C ⊗ [ SU (3) L ⊗ SU (3) c ⊗ U (1) X ]). In this model only exotic techniquarks ( U ′ , D ′ ) will acquire dynamicallygenerated mass due U (1) X interaction at Λ ∼ O ( T eV ) . The terminology exotic refers to3omenclature used in 331 models to designate the allocation of fractional charges assignedto the new quarks (
T, D ). In analogy with this nomenclature ( U ′ , D ′ ) are termed ”exotictechniquarks”, and all techniquarks are in the fundamental representationn, ( R = F ), of SU ( N ) T C .Technicolor models with fermions in the fundamental representation are subject to strongexperimental constraints that comes from the limits on the S parameter. In our case, thecontribution due to the TC sector should still lead to a value to the S parameter compatiblewith the experimental data. At low energies, i.e. at the scale associated with electroweaksymmetry breaking, we should only consider the contribution of four techniquarks because(U ’and D’) are singlets of SU (2) L and do not contribute directly to the bosons (W and Z)massesThe exotic technifermions will present the extreme walking behavior, the usual tech-niquarks ( U, D ) will present the known asymptotic self-energy behavior predicted by theoperator product expansion(OPE)[7]. The exotic techniquarks will have two different en-ergy scales and the 331-TC model corresponds to an example of two-scale Technicolor (TC)model. This article is organized as follows: In section II we present the U (1) X contributionto fermions and technifermions self-energy, in section III we compute the dynamically gener-ated masses to heavy exotic quarks ( T, S, D ) and techniquarks ( U ′ , D ′ ) , where we reproducethe results obtained in Ref.[5] for heavy exotic quarks. In the section IV we illustrate how toobtain the extreme walking behavior in the context of 331-TC model and Section V containsour conclusions. II. THE U (1) X CONTRIBUTION TO FERMIONS AND TECHNIFERMIONSSELF-ENERGY
As described in Refs.[5, 6] the gauge symmetry breaking in 3-3-1 models can be imple-mented dynamically because at the scale of a few TeVs the U (1) X coupling constant becomesstrong. The exotic quark T introduced will form a condensate breaking SU (3) L ⊗ U (1) X to electroweak symmetry, in this paper we will numerically determine the U (1) X contri-bution to dynamic mass of exotic quarks ( T, S, D ) and techniquarks ( U ′ , D ′ ) that appearin the fermionic content of the model[6] following the same procedure described in Ref.[5].The Schwinger-Dyson equation for quarks(techni-quarks) due to U (1) X interaction can be4ritten as[5, 6] S − ( p ) = p − i Z d q (2 π ) Γ µ ( p, q ) S ( q )Γ ν D µν MZ ′ ( p − q ) (2)where in the equation above we assumed the rainbow approximation for the vertex Γ µ,ν ,with Γ µ,ν = ( g V γ µ,ν − g A γ µ,ν γ ), g V = g X ( Y L + Y R ) / g A = g X ( Y L − Y R ) /
4, where Y i are U (1) X hypercharges attributed at chiral components of the exotic quarks(techniquarks).With the purpose of simplifying the calculations it is convenient to choose the LandauGauge. In this case the Z ′ propagator can be written in the following form iD µν MZ ′ ( p − q ) = − i [ g µν − ( p − q ) µ ( p − q ) ν / ( p − q ) ]( p − q ) − M Z ′ . Writing the quark propagator as iS − F ( p ) = i ( /p − Σ X ( p )), and considering the equationabove, we finally can write in the euclidean space the gap equation for Σ( p )[5]Σ X ( p ) = a Z p d qq Σ X ( q )[ q + Σ X ( q )] 1[ p + M Z ′ ] + a Z Λ p d qq Σ X ( q )[ q + Σ X ( q )] 1[ q + M Z ′ ] (3)where a = g X Y L Y R π = βY L Y R . To obtain the last equation we assumed the angle approxima-tion to transform the term p − q ) + M Z ′ as 1( p − q ) + M Z ′ = θ ( p − q ) p + M Z ′ + θ ( q − p ) q + M Z ′ . (4)The integral equation described above can be transformed into a differential equation for f ( x ) introducing the new variables x = p M , with f ( x ) = Σ X ( x ) M and α = M Z ′ M that wereproduce below f ′′ ( x ) + 2 x + α f ′ ( x ) + βY L Y R ( x + α ) xf ( x )( x + f ( x )) = 0 , (5)where M ≡ Σ X (0) is the dynamical mass of exotic quarks(or techniquarks) generated by U (1) X interaction and the respective boundary conditions for f ( x ) are f (0) = 1 and f ′ (0)= 0. In order to obtain the mass spectrum generated due to U (1) X interaction, we followthe same procedure described in Ref.[5] where the ( Y i ) hipercharges of exotic quarks ( T, S,D ) were assumed according to the table 1 and we include the corresponding hipercharges ofexotic techniquarks ( U ′ , D ′ ) [6] that are singlets of SU (3) c .5 ABLE I: ( Y i ) Hipercharges of exotic quarks ( T, S, D ) and exotic techniquarks ( U ′ , D ′ ). Y L Y R Exotic Fermion Charge(Technifermion) − / − / / / / − / − − / − / III. DYNAMICALLY GENERATED MASSES OF EXOTIC QUARKS ANDTECHNIQUARKS DUE U (1) X CONTRIBUTION
As commented in the previous section we follow the same procedure described in Ref.[5],therefore, in order to get an estimate of the U (1) X dynamically generated mass, for exoticfermions(or technifermions), we will numerically solve the Eq.(3) imposing an ultravioletcutoff Λ on this equation. If the gap equation accepts a M T solution( M T is mass of exoticquark (T))[28], then the gauge symmetry SU (3) L ⊗ U (1) X is dynamically broken to SU (2) L ⊗ U (1) Y and the Z ′ , V ± , and U ±± gauge bosons become massive. The Z ′ mass is given by M Z ′ = g X F Π (cid:0) Y TL − Y TR (cid:1) , where F Π is the pseudoscalar decay constant and we calculate itusing the Pagels-Stokar approximation that is given by F Π = 14 π Z dp p ( p + Σ X ( p )) (cid:20) Σ X ( p ) − p d Σ X ( p ) dp Σ X ( p ) (cid:21) (6)Assuming the set of variables described below of Eq.(4), the above equation together withthe definition of M Z ′ , allows to write the Pagels-Stokar relation as1 = βα Z (cid:18) Λ MZ ′ (cid:19) α dxx ( x + f ( x )) (cid:20) f ( x ) − x df ( x ) dx f ( x ) (cid:21) , (7)where the coefficients α and β were defined in the previous section.According to the description given in [5] the consistency requirement imposed about thesolution of Eq.(5) is that the mass of the exchanged particle Z ′ (or α = M Z ′ M ) has to be equalto Z ′ mass (or α ) obtained using Eq.(7). In other words the solutions of the gap equationEq.(5) are iteratively improved by starting with a trial guess for the exchanged boson massand then comparing it with the predicted mass obtained using the Eq.(7). In the Fig I weshow the behavior of Eq.(7) assuming the numerical solution of Eq. (5) for Λ = 42 T eV ,6 ABLE II: The dynamically generated heavy exotic quarks(techniquarks) masses for different U (1) X gauge coupling g X and cutoff Λ.Λ( T eV ) M Z ′ ( T eV ) g X β M T ( T eV ) M S,D ( T eV ) M U ′ ,D ′ ( T eV )42 2.00 2.576 0.0420 5.73 0.76 0.8242 2.50 2.576 0.0420 7.57 0.55 0.6790 3.00 2.541 0.0409 8.22 0.92 1.12162 4.00 2.526 0.0404 10.64 1.22 1.41196 4.50 2.523 0.0403 11.94 1.34 1.47247 5.00 2.514 0.0400 13.16 1.52 1.75 M Z ′ = 2 T eV as a function of parameters α and β . In the table II we show the resultsobtained for the dynamically generated masses for heavy exotic quarks ( T, S, D ) and exotictechniquarks ( U ′ , D ′ ) , where we reproduce the results obtained in Ref.[5] for heavy exoticquarks . IV. THE EXTREME WALKING BEHAVIOR IN A 331-TC MODEL
Theories with large anomalous dimensions ( γ m >
1) are quite desirable for technicolorphenomenology [9, 18], it is known for a long date that four-fermion interactions are re-sponsible for harder self-energy solutions in non-Abelian gauge theories ( γ m ∼ U (1) X interaction only theexotic techniquarks ( U ′ , D ′ ) will acquire an dynamical mass M U ′ = M D ′ = O ( T eV ) atΛ ∼ O ( T eV ). The result is that at this energy scale a bare mass appears in the (TC)Schwinger-Dyson equation assigned to exotic techniquarks, ( U ′ , D ′ ) , what leads to a very”hard” self-energy, or a self-energy of the irregular type, Eq.(1), only for exotic techniquarks( U ′ , D ′ ). In this section considering a four-fermion approximation for U (1) X interaction as-sociated to these techniquarks we will show that the results for M U ′ = M D ′ = O ( T eV ) areof the same order as obtained in the previous section, in this case F Π equation is given by F = 14 π Z dp p M X ( p + M X ) (8)7 IG. 1: Plot of numerical solution of Eqs. (5) and (7) as function of parameters α and β toΛ = 42 T eV , M Z ′ = 2 T eV . To g X = 2 . β = 0 . M T = 5 . T eV ( α = 0 . M T described in Ref.[5]. where M X = M U ′ = M D ′ or M T . As in the previous section, the equation above togetherwith the definition of M Z ′ , allows us to write1 ≈ βα Z dp p ( p + M X ) . (9)For a similar choice of parameters, used in the previous section, for example Λ = 42 TeV, β = 0 .
042 and M Z ′ = 2 . M T ≈ . M U ′ = M D ′ ≈ . U (1) X to the mass of exotic techniquarks can be approximated by afour-fermion interaction and the exotic techniquarks exhibit a self-energy behavior of theirregular type.To illustrated the extreme walking behavior exhibited only by the exotic technifermionswe consider the full gap equation for the ”exotic techniquark U’” that contains the sum oftwo contributions, the U (1) X interaction and TC interaction, and we consider the presenceof dynamically massive technigluons. The problems for chiral symmetry breaking in this casehave been discussed recently, where confinement may play an important role[21, 23, 24]. In8his work we consider that technigluons acquire a dynamical mass along the line of QCDas proposed by Cornwall[22] many years ago and the dynamical technigluon mass behavesas[23, 24] m tg ( k ) = m tg (0) k + m tg (0) , (10)with m tg (0) ≈ Λ T C and the technifermion dynamical mass will be given by M T C ( p ) = C (2 π ) Z K ( g, p ) d k [ k + M T C ( k )] , (11)where K ( g, p ) = 3¯ g tc ( p − k ) M T C ( k )( p − k ) + m tg (( p − k ) ) M T C ( p ) is the dynamical techniquark mass, and C is the Casimir operator for the tech-nifermionic representation with effective TC coupling ¯ g tc , given by¯ g tg ( p ) = 1 b T C ln[( p + 4 m tg ( p )) / Λ T C ] , (12)in this expression b T C is the first β T C function coefficient and we consider the Landaugauge. Similarly to the previous section the integral equation described by Eq.(11) can betransformed into a differential equation for h ( x ) introducing the new variables x = p M TC (0) ,with h ( x ) = M TC ( x ) M TC (0) and δ = m tg (0) M TC (0) we obtain h ′′ ( x ) + 2 x + δ h ′ ( x ) + 3 C g T C ( x )16 π ( x + δ ) xh ( x )( x + h ( x )) = 0 . (13)The full gap equation for the ”exotic techniquark U’” can be written as M U ′ ( x ) = M T C (0) h ( x ) + M X (4 f ) , (14)in the Fig.(2a)[Blue curve] we show the behavior of M T C ( x ) = h ¯ UU i Λ , that corresponds to thedynamical mass generated for the the usual techniquarks ( U, D ) obtained with Eq.(13). Inthe Fig.(2b)[Red curve] we include the contribution of U (1) X effective four-fermion interac-tion and we show the behavior of the dynamical mass generated for the exotic techniquarks( U ′ , D ′ ), which is described by M U ′ ( x ) = h ¯ U ′ U ′ i Λ .9
00 10000,11 m U’ = / m U =
In the Refs.[14–16, 25] we discussed the possibility of obtaining a light composite TCscalar boson, this result is a direct consequence of extreme walking (or quasi-conformal)technicolor theories, where the asymptotic self-energy behavior is described by Eq.(1). Theextreme walking technicolor can be obtained in three different ways and in this paper wepropose an scheme to obtain the quasi-conformal behavior based on an extension of theelectroweak sector of the standard model, in the so called 331-TC model ( SU ( N ) T C ⊗ SU (3) L ⊗ SU (3) c ⊗ U (1) X ), where the exotic quark T introduced will form a U (1) X condensate( h ¯ T T i ) breaking SU (3) L ⊗ U (1) X to electroweak symmetry that is broken by an usualtechnifermion condensate. As the comment presented in the introductory section, 3-3-1models predicted new massive weak gauge bosons, Z ′ and V ± , and this model has thenecessary requirements to explain the reported diboson anomaly.10ollowing the same procedure described in Ref.[5], the solutions of the gap equation,Eq.(5), were iteratively improved by starting with a trial guess for the exchanged bosonmass and then comparing it with the predicted mass obtained using the Eq.(7). In thetable II we show the results obtained for the dynamically generated masses of heavy exoticquarks ( T, S, D ) and exotic techniquarks ( U ′ , D ′ ) , where we reproduce the results obtainedin Ref.[5] for heavy exotic quarks. In Ref.[26] we discuss a mechanism for the dynamical massgeneration, including the mass generation for the t quark, in the case of grand unified theoriesthat incorporate quarks and techniquarks. We expect that a similar mechanism to the onedescribed in [26] can be developed and incorporated in the present model. In the section 4we show that the results obtained in the table II, ( M U ′ = M D ′ = O ( T eV )), are of the sameorder as the ones obtained with a four-fermion approximation for the U (1) X interactionand in the Fig.2 [Red Curve ] we show the extreme walking behavior displayed only byexotic techniquarks ( U ′ , D ′ ) due to their strong U (1) X interaction because Y U ′ L Y U ′ R = 3(see table I). The exotic technifermions have two different energy scales and the 331-TCmodel corresponds to an example of two-scale Technicolor (TC) model, in the Ref.[16] weconsidered the possibility of a light TC composite boson arising from mass mixing betweena relatively light and heavy composite scalar singlets from a see-saw mechanism expected tooccur in two-scale TC model. We emphasize that the see-saw mechanism expected to occurin this model is not exactly the same one described in the Ref[16], in the model proposedthe extreme walking behavior displayed by exotic techniquarks is due to their strong U (1) X interaction and in this case in Eq.(1), µ ≈ Λ = Λ , and Λ it is not generated by the TCsector. In order to provide a example we consider the SU (2) T C case with the technifermioniccontent showed in[6], where n T F = 6. With this we obtain m φ ∼
124 GeV, m φ ∼ . = 3 T eV and Λ
ET C ≈ Acknowledgments
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1, considering that α c (Λ X ) is close to 1, we can roughly estimate that U (1) X condensationshould occur only for the channel where ( Y L Y R ) ≥
1. Once ( Y L Y R = ) the quark T channelis the one leading to the most attractive channel) the quark T channelis the one leading to the most attractive channel